Question

use technology to find the indicated area under the standard normal curve. include an appropriately labeled sketch of the normal curve and shade the appropriate region.
a. find the probability that a z - score will be 0.91 or less.
b. find the probability that a z - score will be 0.91 or more.
c. find the probability that a z - score will be between - 1.4 and - 1.04
the probability that a z - score will be 0.91 or more is 0.1814
(round to four decimal places as needed.)
c. which graph below shows the probability that a z - score is between - 1.4 and - 1.04?
○a.
○b.
○c.
○d.

Explanation:

Step1: Recall the properties of the standard normal distribution

The standard normal distribution is symmetric about \(z = 0\). The total area under the standard normal curve is \(1\). We can use a standard normal table (z - table) or a calculator with a normal - distribution function (e.g., the normalcdf function on a TI - 84 Plus calculator).

Step2: Calculate \(P(Z\leq0.91)\)

Using a standard normal table or a calculator (for a TI - 84 Plus, we can use the normalcdf function with lower bound \(-\infty\) (represented as a very large negative number, say \(-1000\)) and upper bound \(0.91\), mean \(\mu = 0\) and standard deviation \(\sigma=1\)).
\(P(Z\leq0.91)=\Phi(0.91)\)
Using a standard normal table or calculator, \(P(Z\leq0.91)\approx0.8186\)

Step3: Calculate \(P(Z\geq0.91)\)

Since \(P(Z\geq z)=1 - P(Z < z)\) (by the complement rule of probability).
We know from Step 2 that \(P(Z < 0.91)\approx0.8186\)
So \(P(Z\geq0.91)=1 - 0.8186 = 0.1814\)

Step4: Calculate \(P(-1.4<Z<-1.04)\)

Using the formula \(P(a < Z < b)=\Phi(b)-\Phi(a)\)
For \(a=-1.4\) and \(b = - 1.04\)
\(\Phi(-1.4)\) (the area to the left of \(z=-1.4\)) and \(\Phi(-1.04)\) (the area to the left of \(z=-1.04\))
Using a standard normal table or calculator (normalcdf(-1.4,-1.04,0,1)):
\(\Phi(-1.4)=0.0808\), \(\Phi(-1.04)=0.1492\)
\(P(-1.4 < Z < -1.04)=\Phi(-1.04)-\Phi(-1.4)=0.1492 - 0.0808=0.0684\)

For part c, when we want to find the probability that \(z\) is between \(-1.4\) and \(-1.04\), we are looking for the area between two negative \(z\) - scores. The graph should have the shaded region between \(z=-1.4\) and \(z=-1.04\).

Answer:

a. \(0.8186\)
b. \(0.1814\)
c. \(0.0684\)